A short proof of Stein's universal multiplier theorem

We give a short proof of Stein's universal multiplier theorem, purely by probabilistic methods, thus avoiding any use of harmonic analysis techniques (complex interpolation or transference methods)

Bi-stochastic kernels via asymmetric affinity functions

In this short letter we present the construction of a bi-stochastic kernel p for an arbitrary data set X that is derived from an asymmetric affinity function {\alpha}. The affinity function {\alpha} measures the similarity between points in X and some reference set Y. Unlike other methods that construct bi-stochastic kernels via some convergent iteration process or through solving an optimization problem, the construction presented here is quite simple. Furthermore, it can be viewed through the lens of out of sample extensions, making it useful for massive data sets.Comment: 5 pages. v2: Expanded upon the first paragraph of subsection 2.1. v3: Minor changes and edits. v4: Edited comments and added DO

LpL^p estimates for the Hilbert transforms along a one-variable vector field

Stein conjectured that the Hilbert transform in the direction of a vector field is bounded on, say, $L^2$ whenever $v$ is Lipschitz. We establish a wide range of $L^p$ estimates for this operator when $v$ is a measurable, non-vanishing, one-variable vector field in \bbr ^2. Aside from an $L^2$ estimate following from a simple trick with Carleson's theorem, these estimates were unknown previously. This paper is closely related to a recent paper of the first author (\cite{B2}).Comment: 25 page

Estimates for compositions of maximal operators with singular integrals

We prove weak-type (1,1) estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $\Delta^*\Psi$ where $\Delta^*$ is Bourgain's maximal multiplier operator and $\Psi$ is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the $L^q$ operator norm when $1 < q < 2$. We also consider associated variation-norm estimates

Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group: an expanded version

Marcinkiewicz multipliers are L^{p} bounded for 1<p<\infty on the Heisenberg group H^{n}\simeqC^{n}\timesR (D. Muller, F. Ricci and E. M. Stein) despite the lack of a two parameter group of automorphic dilations on H^{n}. This lack of dilations underlies the inability of classical one or two parameter Hardy space theory to handle Marcinkiewicz multipliers on H^{n} when 0<p\leq1. We address this deficiency by developing a theory of flag Hardy spaces H_{flag}^{p} on the Heisenberg group, 0<p\leq1, that is in a sense `intermediate' between the classical Hardy spaces H^{p} and the product Hardy spaces H_{product}^{p} on C^{n}\timesR. We show that flag singular integral operators, which include the aforementioned Marcinkiewicz multipliers, are bounded on H_{flag}^{p}, as well as from H_{flag}^{p} to L^{p}, for 0<p\leq1. We characterize the dual spaces of H_{flag}^{1} and H_{flag}^{p}, and establish a Calder\'on-Zygmund decomposition that yields standard interpolation theorems for the flag Hardy spaces H_{flag}^{p}. In particular, this recovers the L^{p} results by interpolating between those for H_{flag}^{p} and L^{2} (but regularity sharpness is lost).Comment: At 113 pages, this is an expanded version of the paper that includes much detai